General Instructions

To do each example, just click the Submit button. You do not have to type in any R instructions or specify a dataset. That's already done for you.

The Wilcoxon Rank Sum Test

Example 4.1 in Hollander and Wolfe.



The Associated Point Estimate (Median of the Pairwise Differences)

The Hodges-Lehmann estimator associated with the rank sum test is the median of the pairwise differences, which are the nx ny differences

YjXi,     for all i and j

Example 4.3 in Hollander and Wolfe.


The Associated Confidence Interval

Very similar to the confidence intervals associated with the sign test and signed rank test, the confidence interval has the form

(D(k), D(m + 1 − k))

where m = nx ny is the number of pairwise differences, the Di are the pairwise differences, and, as always, parentheses on subscripts indicates order statistics. That is, one counts in k from each end in the list of sorted pairwise differences to find the confidence interval.

Example 4.4 in Hollander and Wolfe.



The R Function wilcox.test

All of the above can be done in one shot with the R function wilcox.test (on-line help).

Only one complaint. It does not report the actual achieved confidence level (here 96.0%) but rather the confidence level asked for (here 95%, the default). If you want to know the actual achieved confidence level, you'll have to use the code in the confidence interval section above. But you can use wilcox.test as a convenient check (the intervals should agree).

Warning About Ties and Zeros

Do not use the wilcox.test function when there are ties or zeros in the data. See the following section.

The R Function wilcox.exact

There is an R function wilcox.exact (on-line help) that does do hypothesis tests correctly in the presence of ties.

It does not do confidence intervals or point estimates correctly in the presence of ties. Use the code in the confidence interval section or the point estimate section above.

In order to see what's going on, let's copy some of the code from the beginning of the calculation without the function. This shows the ranks so we see the tied ranks and shows the calculation of the test statistic u so we can see that it agrees with the test statistic calculated by wilcox.exact.

Fuzzy Procedures

These are analogous to the fuzzy procedures for the sign test explained on the sign test and related procedures page and on the fuzzy confidence intervals and P-values page.

Since they are so similar, we won't belabor the issues and interpretations. The only difference is that for fuzzy confidence intervals the jumps in the plot are at YX differences (no surprise) rather than at order statistics and for fuzzy P-values they are at numbers in the CDF table for the null distribution of the test statistic, which is now the Mann-Whitney distribution rather than the symmetric binomial distribution.

In short, the distributions have changed but everything else remains the same.

Fuzzy P-Values

The fuzzy P-value is (not very uniformly) distributed over the interval from 0.0041 to 0.0175. This is fairly strong evidence against the null hypothesis.

The only virtue this procedure has over the wilcox.exact procedure illustrated in the preceding section is that this procedure is exact at all significance levels, whereas the wilcox.exact only gives an exact procedure for the significance levels that appear in the CDF table of the null distribution of the test statistic (which is not tabulated anywhere, just calculated by wilcox.exact, since the presence of tied ranks changes the distribution, so it is not the distribution calculated by pwilcox or tabulated in the textbook).

Both procedures are exact in some sense (not exactly the same sense). Both say more or less the same thing. Certainly, fairly strong evidence against the null hypothesis is what both say.

You can use whichever you like. What you should not do is follow traditional procedures, described by the textbook and implemented in wilcox.test, when there are ties.